The purpose of this activity is to find multiples of two different numbers in context. Students decide possible table sizes for a party based on whether or not a given number of people is a multiple of 6, 8, both, or neither. In situations where a given number is not a multiple of 6 or 8, they reason about what it means in context (MP2).

Focus the Activity Synthesis on how the number of seats at the two table sizes relate to multiplication and what it means when the number of people is not a multiple of the number of seats at a table.

• Invite students who used drawings, the idea of multiples, or multiplication facts to share while discussing the following questions.
• “Can Noah’s class fit exactly in tables for 6? How do you know?” (Yes. , so Noah’s class will fill 5 tables exactly.)
• “Can Noah’s class fit exactly in tables for 8? How do you know?” (No, 30 is not a multiple of 8. They would need at least 4 tables but that’s too many seats, so 2 of the seats would be empty.)
• “How do multiplication facts or the idea of multiples help us think about this problem?” (We know that 24 and 32 are products of 8 and another number, but 30 is not.)
• “For the problem with both classes and the teachers, could they all fit at tables of 6? Why or why not?” (Yes, they could get 9 tables of 6, but there will be 4 empty seats. If they don’t want empty seats, tables of 6 won’t work because 50 is not a multiple of 6.)
• “Could they all fit at tables of 8? Why or why not?” (Yes, they could get 7 tables of 8, but there will be 6 empty seats. If they don't want empty seats, tables of 8 won't work because 50 is not a multiple of 8.)

In this activity, students solve problems that involve finding multiples that are shared by two different numbers: the number of party hats in a package and the number of noise-makers in a package. They reason about how many of each package to get so that a certain number of children receive one party hat and one noise-maker. As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).

To solve the problems, students may decontextualize the situation and reason about factors and multiples, and then recontextualize the solutions in terms of each person receiving a party hat and noise-maker. As they do so, they practice reasoning quantitatively and abstractly (MP2). During the Activity Synthesis, analyze different solutions and discuss why numbers that are multiples both of 8 and 10 are useful in this situation.

• “How are you choosing which numbers to try?”
• “How can you use multiples to solve the problem?”

• Invite students to share their responses to the problem about 72 party hats, including those who get 70 party hats and 72 noise-makers or 80 party hats and 80 noise-makers. Encourage students to highlight how they are using multiples in their reasoning.
• “Why are there different answers to this question?” (The numbers don’t work out exactly to get 72 party hats and 72 noise-makers, so we need to look at different alternatives.)
• “Is it possible to buy exactly the same number of party hats and noise-makers? If so, how many of each?” (Yes, 40 or 80.)
• “Why is it possible to buy exactly the same number of party hats and noise-makers when the packaging is different?” (Some multiples of 10 are also multiples of 8, such as 40 and 80.)